It has already been a few years or so ago now since the editors of
a somewhat obscure magazine asked me to contribute an article on the
question whether or not computers could think. I did not feel like doing
that and I explained my refusal with the remark that I found the
suggested topic just as unimportant as the similarly burning question
whether or not submarines could swim. (I had reckoned without my host: the
editor —a sociologist— wrote me back, that he found that last
question very interesting as well!)

The question of whether or not computers can think is of no use, because
anyone who tries to answer this, is very soon on a one-way trip to
philosophy. The question is also redundant in the sense that we can
think about computers —their capabilities, their limitations and
their impact on culture— without having to choose what we should,
could or want the word "think" to mean.

I do not need to itemize the mystical nonsense, ranging from
electronic brains to monsters of Frankenstein, with which popular
belief and the tabloids surrounded computers, because that is well
known. Much more important is the phenomenon of the abundance and
the persistence of this nonsense, because as a symptom this
phenomenon is explainable in only one way: it tells us that
computers were, when they appeared, a radical novelty that could not
be adequately understood in terms of already familiar phenomena and
concepts. Comparisons with things of the past were all ill-suited,
analogies to what we knew proved time and time again too superficial
to be not misleading.

The radical novelty had two aspects. The best known of them was
the incredible speed: computers calculated first a thousand times
faster than previously possible, then a million, and finally a
billion times faster. The problem is that such speed ratios,
although easy to write, are in a very real sense, unimaginable, and
it is very difficult to imagine how unimaginably unimaginable something is.

Even a single factor of ten is like night and
day; once I asked a lady friend, to convince her of that, how many children
she had — I knew there were six and she understood what I meant. And
a baby who learns to crawl a thousand times as quickly will overtake
fighter jets! For the sake of completeness I mention that also the
capacity for information storage —in anthropomorphic terminology
also called "memory size"— has increased unimaginably over the
years. In short: at such drastic quantitative differences all
analogies fail.

The other aspect of the radical novelty initially appeared
slightly less in the foreground, but the repercussions of it are no
less important. To get this aspect in the spotlight I must allow
myself some poetic license —or mathematical abstraction— and I hope
that the reader allows me. I want to refer to "formulae" which are
constructed from "symbols" from a finite "alphabet". That "alphabet"
is not limited to the 26 letters, we allow punctuation, all sorts of
brackets and mathematical symbols and the digits 0 through 9 as
well. The advantage of this poetic license is that it allows us to
put an algebraic expression like (a+b)/c, a program fragment like x :=
x+1, and a decimal number like 729 all three under the same heading
"formula".

The just outlined imagery enables me to indicate the second
novelty: with computers, a universe is created where nothing else
happens than deriving new formulae from existing ones, according to
strict rules. This universe is theoretically of deep meaning because
everything that is derivable can be derived in this way; the
electronic version is of great practical significance because it
works so fast that even the result of long derivations can become
available in an acceptable time frame.

Such a formal universe is therefore as novelty radical because it
blatantly goes against all our previously acquired intuitions.
Because e.g. all formulae are constructed from the symbols of a
finite alphabet, there is no such thing as continuous change. We
could try to talk about "small" changes —e.g. the change of just one
single symbol— but again our intuition fails us, because no metrics
exist according to which "small" changes also have "small" effects.

The most salient feature of the formal universe is, however, that
nothing other than formula manipulation takes place; even the
smallest fraction of a verbal argument is thus excluded. Insofar as
we identify "understanding" and "insight" with verbal reasoning,
understanding and insight are therefore excluded from the formal
universe. A small example may clarify this: if we need to establish
in the framework of a larger discussion that 812 is a multiple of 7,
then that is not something that we try to "understand" or "see", but
something that we determine by just performing a simple long
division, by —in our imagery— doing some formula manipulation.

Our traditional ways of arguing are mixed, viz. partly formal and partly
verbal, and verbal reasoning is so strongly taught at an early age
that total elimination of the verbal component appears to many at
first sight unwanted, if not impossible. Further analysis has shown
however, that it is always the verbal component that is vague,
unintelligible and ambiguous. Traditionally, mathematics has
limited itself to, for otherwise too subtle reasoning, reducing the
verbal component by partial formalization; the formal universe, such
as that embodied by computers, is challenge and incentive to go much
further, until in the end, the verbal component is completely
eliminated. This challenge has not remained unanswered, and the
result is that in the area of mathematical methodology a quiet
revolution is taking place in which full formalization increasingly
becomes attractive.

How "important" is this all? Who initially sees science as an
activity where only a negligible fraction of the population
participates actively, and then sees the science as a dispersed
whole in which the mathematical subculture has maneuvered itself
into a kind of intellectual ghetto, will at best mumble "well, well,
very interesting", shrug and get back to the usual daily routine.

To begin with, one can also consider science as one of the main
pillars of our culture and then —one would not be the first!— argue
that intellectual standing and weight of each scientific discipline
is determined by its mathematical content. In that vision, a radical
change of course in mathematics would leave in the long term
footprints in the vast majority of our intellectual life.

I expect such a radical change of course. At this moment, the
Concise Oxford Dictionary still defines mathematics as "abstract
science of space, number, and quantity"; when the aforementioned quiet
revolution has taken place, "art and science of effective reasoning"
will be a more adequate definition of mathematics. I will not be around
to see it happen, but with a bit of luck, in a hundred years
from now, the question of whether or not computers can think will be no
more than a historical curiosity from the twentieth century.